The family of deterministic input-driven pushdown automata (IDPDA; a.k.a. visibly pushdown automata, a.k.a. nested word automata) is known to be closed under reversal, concatenation and Kleene star. As shown by Alur and Madhusudan ("Visibly pushdown languages", STOC 2004), the reversal and the Kleene star of an n-state IDPDA can be represented by an IDPDA 2^O(n2) with states, while concatenation of an m-state and an n-state IDPDA is represented by an IDPDA with 2^O((m+n)2) states. This paper presents more efficient constructions for the reversal and for the Kleene star, which yield 2 ^{Θ(n log n)} states, as well as an m 2 ^{Θ(n log n)}-state construction for the concatenation. These constructions are optimal due to the previously known matching lower bounds.